[Solution] (a) Starting with the geometric series ∑_n=0^∞ x^n, find the sum of the series ∑_n=1^∞ n x^n-1 |x|<1 (b) Find the sum of
Question: (a) Starting with the geometric series \(\sum_{n=0}^{\infty} x^{n}\), find the sum of the series
\[\sum_{n=1}^{\infty} n x^{n-1} \quad|x|<1\](b) Find the sum of each of the following series.
- \(\sum_{n=1}^{\infty} n x^{n}, \quad|x|<1\)
- \(\sum_{n=1}^{\infty} \frac{n}{2^{n}}\)
(c) Find the sum of each of the following series.
- \(\sum_{n=2}^{\infty} n(n-1) x^{n},|x|<1\)
- \(\sum_{n=2}^{\infty} \frac{n^{2}-n}{2^{n}}\)
- \(\sum_{n=1}^{\infty} \frac{n^{2}}{2^{n}}\)
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![[Solution Library] (a) By completing the square,](/images/solutions/MC-solution-library-60232.jpg "[Solution Library] (a) By completing the square, show that ∫_0^1")
[Solution Library] (a) By completing the square, show that ∫_0^1
(See Steps) Find the Maclaurin series for f(x) using the definition
Solution: Find the Maclaurin series for f(x) using the definition
![[Solution] List the first five terms of](/images/solutions/MC-solution-library-60235.jpg "[Solution] List the first five terms of the sequence. 2 •")
[Solution] List the first five terms of the sequence. 2 •
Solution: Determine whether the geometric series is convergent
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